Among the well known nondestructive testing techniques is the technique of spectroscopic ellipsometry, which measures reflectance data by reflecting electromagnetic radiation from a sample (typically to measure the thickness of a very thin film on a substrate). In spectroscopic ellipsometry, an incident radiation beam having a known polarization state reflects from a sample, and the polarization of the reflected radiation is analyzed to determine properties of the sample. Since the incident radiation includes multiple frequency components, a spectrum of measured data (including data for incident radiation of each of at least two frequencies) can be measured. Typically, the polarization of the incident beam has a time-varying characteristic (produced, for example, by passing the incident beam through a mechanically rotating polarizer), and/or the means for analyzing the reflected radiation has a time-varying characteristic (for example, it may include a mechanically rotating analyzer). Examples of spectroscopic ellipsometry systems are described in U.S. Pat. No. 5,329,357, issued Jul. 12, 1994 to Bernoux, et al., and U.S. Pat. No. 5,166,752, issued Nov. 24, 1992 to Spanier, et al. Spectroscopic ellipsometry theory is described in F. Bernoux, et al., "Ellipsometrie," Techniques de l'Ingenieur, R6490, pp. 1-16 (1990).
Reflectance data (measured by spectroscopic ellipsometry or other reflection techniques) are useful for a variety of industrial applications. The thickness of various coatings (either single layer or multiple layer) on a substrate can be determined from spectroscopic ellipsometry data (indicative of the polarization of radiation reflected from the sample in response to incident radiation having known polarization state), or a reflectance spectrum or relative reflectance spectrum.
The reflectance of a sample (or sample layer) at a single wavelength can be determined by analyzing spectroscopic ellipsometry data (indicative of the polarization changes of radiation reflected from the sample, in response to incident radiation having known polarization state) or extracted from an accurately measured reflectance or relative reflectance spectrum. It is useful to determine sample reflectance in this way where the reflectance of photoresist coated wafers at the wavelength of a lithographic exposure tool must be found, to determine proper exposure levels for the wafers or to optimize the thickness of the resist to minimize reflectance of the entire coating stack.
The present invention pertains to calibration of an ellipsometer (such as a spectroscopic ellipsometer). To appreciate the difference between the inventive calibration method, and conventional calibration methods, it is helpful to consider the method of operation of a spectroscopic ellipsometer.
FIG. 1 is a schematic diagram of a typical spectroscopic ellipsometer. In operation of this ellipsometer, a beam of broadband radiation from broadband radiation source 150 is linearly polarized in polarizer 152, and the linearly polarized beam is then incident on sample 154o After reflection from sample 154, the beam propagates toward analyzer 156 with a changed polarization state (typically, the reflected beam has elliptical polarization, where the polarized beam emerging from polarizer 152 had linear polarization). The reflected beam propagates through analyzer 156 into dispersion element (spectrometer) 158. In dispersion element 158, the beam components having different wavelengths are refracted in different directions to different detectors of detector array 160. Processor 162 receives the measured data from each detector of array 160, and is programmed with software for processing the data it receives in an appropriate manner. Detector array 160 can be a linear array of photodiodes, with each photodiode measuring radiation in a different wavelength range.
Either polarizer 152 or analyzer 156 is rotatably mounted for rotation about the optical axis during a measurement operation (or both of them are so rotatably mounted). During a typical measurement operation, polarizer 152 is rotated and analyzer 156 remains in a fixed orientation, or analyzer 156 is rotated and polarizer 152 remains fixed.
Processor 162 can be programmed to generate control signals for controlling the rotation (or angular orientation) of polarizer 152 and/or analyzer 156, or for controlling other operating parameters of elements of the FIG. 1 system (such as the position of a movable sample stage on which sample 154 rests). Processor 162 can also receive data (indicative of the angular orientation of analyzer 156) from an analyzer position sensor associated with analyzer 156 and data (indicative of the angular orientation of polarizer 152) from a polarizer position sensor associated with polarizer 152, and can be programmed with software for processing such orientation data in an appropriate manner.
If polarizer 152 is controlled so that it rotates at a constant speed, the signal received at each detector of array 160 will be a time-varying intensity given by: ##EQU1## where I.sub.0 is a constant that depends on the intensity of radiation emitted by source 150, .omega. is the angular velocity of polarizer 152, P.sub.0 is the angle between the optical axis of polarizer 152 and the plane of incidence (e.g., the plane of FIG. 1) at an initial time (t=0), and .alpha. and .beta. are sample related values defined as follows: EQU .alpha.=[tan.sup.2 .psi.-tan.sup.2 (A-A.sub.0)]/[tan.sup.2 .psi.+tan.sup.2 (A-A.sub.0)] (2)
and EQU .beta.=2(tan.psi.)(cos.DELTA.)(tan(A-A.sub.0)/[tan.sup.2 .psi.+tan.sup.2 (A-A.sub.0)] (3)
where tan.psi. is the amplitude of the complex ratio of the p and s components of the reflectivity of the sample, .DELTA. is the phase of the complex ratio of the p and s components of the reflectivity of the sample (where "p" denotes the component for polarized radiation whose electrical field is in the plane of FIG. 1, and "s" denotes the component for polarized radiation whose electrical field is perpendicular to the plane of FIG. 1), A is the nominal analyzer angle (a reading of analyzer 156's orientation angle, supplied for example from the above-mentioned analyzer position sensor associated with analyzer 156), and A.sub.0 is the offset of the actual orientation angle of analyzer 156 from the reading "A" (due to mechanical misalignment, A.sub.0 can be non-zero).
The values .alpha.' and .beta.' are also sample related values, defined as follows: EQU .alpha.'=.alpha.cos (2P.sub.0)+.beta.sin (2P.sub.0) (4)
and EQU .beta.'=.alpha.sin (2P.sub.0)-.beta.cos (2P.sub.0) (5)
where .alpha., .beta., and P.sub.0 are defined above.
To achieve measurement accuracy, it is crucial to determine P.sub.0 and A.sub.0 very precisely.
Conventionally, P.sub.0 and A.sub.0 are calibrated simultaneously by a method known as the "minimum residual method," first proposed in David E. Aspnes and A. A. Studna, "High Precision Scanning Ellipsometer," Applied Optics, Vol. 14, No. 1, pp. 220-228 (1975). The minimum residual method is still widely used by ellipsometer users and manufacturers as of the filing date of this specification.
The conventional minimum residual method determines (from measured data) a quantity known as the "residual" (R), which is: EQU R=1-.alpha..sup.2 -.beta..sup.2 ( 6)
where .alpha. and .beta. are defined in equations (2) and (3).
An equivalent quantity is R'=1-.alpha.'.sup.2 -.beta.'.sup.2 where .alpha.' and .beta.' are defined in equations (4) and (5). Of course, it follows from equations (2) through (5) that R=R', and both R and R' are denoted herein as the "residual."
Using equations (2) and (3), it is apparent that the residual, R, can also be expressed as: EQU R=(4-cos.sup.2 .DELTA.) tan.sup.2 .psi. tan.sup.2 (A-A.sub.0)/[tan.sup.2 .psi.+ tan.sup.2 (A-A.sub.0)].sup.2 ( 7)
By orienting the analyzer so that A-A.sub.0 =.delta.A is very small, it can be assumed that tan.delta.A is approximately equal to .delta.A. Under this condition, equation (7) can be approximated by: EQU R=(4-cos.sup.2 .DELTA.) (.delta.A/tan.psi.).sup.2 [1-2(.delta.A/tan.psi.).sup.2 ] (8)
The "phase" of the residual R is defined by: ##EQU2## where .alpha.' and .beta.' are defined by equations (4) and (5).
To perform calibration (i.e., determine the values A.sub.0 and P.sub.0) in accordance with the conventional minimum residual method, the orientation of analyzer 156 is first scanned around the zero position, the values R and "Phase" are determined from the measured data at each measured value (A) of the analyzer's orientation, and the values R and "Phase" are plotted as a function of A, to generate a graph such as that shown in FIG. 2.
Then, the value A.sub.0 (the offset between analyzer 156's actual orientation angle and each reading "A") is identified as the minimum of the "R v. A" curve. When A=A.sub.0, it is true that .alpha.=1 and .beta.=0, so that Phase=2P.sub.0. Thus, having identified the value A.sub.0, the minimum residual method identifies 2P.sub.0 as the value of the "Phase v. A" curve at A=A.sub.0.
To accurately determine the minimum of the "R v. A" curve, it is necessary to fit the bottom part of this curve with a parabolic curve. However, this cannot be done accurately under all conditions, for the reasons explained with reference to FIG. 3.
FIG. 3 is a graph of three "R v. A" curves, each generated from measurements at cos (.DELTA.)=1 a different value of tan(.psi.), namely tan(.psi.)=10, tan(.psi.)=0.1 and tan(.psi.)=1. From FIG. 3, it is apparent that the shape of the "R v, A" curve strongly depends on the value of tan(.psi.). If tan(.psi.) is too small or too large, the bottom part of the curve is very flat, in which case a small perturbation caused by noise can cause a large shift in the estimated value of the curve's minimum position. Thus, the conventional "minimum residual method" is reliable and accurate only for samples for which the value of tan(.psi.) is in a very limited range.
There are several other important limitations of prior art calibration methods (including the "minimum residual method"), including that they can be performed accurately only on very thick samples. Until the present invention, it had not been known how to avoid these limitations of prior art calibration methods.